The General Theory of Dirichlet's Series
By G. H. Hardy and Marcel Riesz
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Following an introduction, the authors proceed to a discussion of the elementary theory of the convergence of Dirichlet's series, followed by a look at the formula for the sum of the coefficients of a Dirichlet's series in terms of the order of the function represented by the series. They continue with an examination of the summation of series by typical means and of general arithmetic theorems concerning typical means. After a survey of Abelian and Tauberian theorems and of further developments of the theory of functions represented by Dirichlet's series, the text concludes with an exploration of the multiplication of Dirichlet's series.
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The General Theory of Dirichlet's Series - G. H. Hardy
The General Theory of Dirichlet's Series
G. H. Hardy
Marcel Riesz
DOVER PHOENIX EDITIONS
Bibliographical Note
This Dover edition, first published in 2005, is an unabridged republication of the work originally published in 1915 by the Cambridge University Press, Cambridge, U.K. It was No. 18 in the series Cambridge Tracts in Mathematics and Mathematical Physics.
Library of Congress Cataloging-in-Publication Data
Hardy, G. H. (Godfrey Harold), 1877-1947.
The general theory of Dirichlet’s series / G.H. Hardy and Marcel Riesz. p. cm.—(Dover phoenix editions)
Originally published: Cambridge: Cambridge University Press, 1915, in series: Cambridge tracts in mathematics and mathematical physics.
Includes bibliographical references.
9780486155173
1. Dirichlet’s series. 2. Number theory. I. Riesz, Marcel, b. 1886. II. Title.
III. Series.
QA295.H3 2005
515’.243—dc22
2005049741
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501
MATHEMATICS QUOTQUOT UBIQUE SUNT
OPERUM SOCIETATEM NUNC DIREMPTAM
MOX UT OPTARE LICET REDINTEGRATURIS
D.D.D. AUCTORES
HOSTES IDEMQUE AMICI
PREFACE
THE publication of this tract has been delayed by a variety of causes, and I am now compelled to issue it without Dr Riesz’s help in the final correction of the proofs. This has at any rate one advantage, that it gives me the opportunity of saying how conscious I am that whatever value it possesses is due mainly to his contributions to it, and in particular to the fact that it contains the first systematic account of his beautiful theory of the summation of series by ‘typical means’.
The task of condensing any account of so extensive a theory into the compass of one of these tracts has proved an exceedingly difficult one. Many important theorems are stated without proof, and many details are left to the reader. I believe, however, that our account is full enough to serve as a guide to other mathematicians researching in this and allied subjects. Such readers will be familiar with Landau’s Handbuch der Lehre von der Verteilung der Primzahlen, and will hardly need to be told how much we, in common with all other investigators in this field, owe to the writings and to the personal encouragement of its author.
G. H. H.
19 May 1915.
Table of Contents
Title Page
Copyright Page
PREFACE
I - INTRODUCTION
II - ELEMENTARY THEORY OF THE CONVERGENCE OF DIRICHLET’S SERIES
III - THE FORMULA FOR THE SUM OF THE COEFFICIENTS OF A DIRICHLET’S SERIES: THE ORDER OF THE FUNCTION REPRESENTED BY THE SERIES
IV - THE SUMMATION OF SERIES BY TYPICAL MEANS
V - GENERAL ARITHMETIC THEOREMS CONCERNING TYPICAL MEANS
VI - ABELIAN AND TAUBERIAN THEOREMS
VII - FURTHER DEVELOPMENTS OF THE THEORY OF FUNCTIONS REPRESENTED BY DIRICHLET’S SERIES
VIII - THE MULTIPLICATION OF DIRICHLET’S SERIES
BIBLIOGRAPHY
DOVER PHOENIX EDITIONS
I
INTRODUCTION
1. The series whose theory forms the subject of this tract are of the form
(1),
where (λn) is a sequence of real increasing numbers whose limit is infinity, and s = σ + ti is a complex variable whose real and imaginary parts are σ and t. Such a series is called a Dirichlet’s series of type λn. If λn = n, then f(s) is a power series in e−s. If λn = log n, then
(2)
is called an ordinary Dirichlet’s series.
Dirichlet’s series were, as their name implies, first introduced into analysis by Dirichlet, primarily with a view to applications in the theory of numbers. A number of important theorems concerning them were proved by Dedekind, and incorporated by him in his later editions of Dirichlet’s Vorlesungen über Zahlentheorie. Dirichlet and Dedekind, however, considered only real values of the variable s. The first theorems involving complex values of s are due to Jensen¹, who determined the nature of the region of convergence of the general series (1); and the first attempt to construct a systematic theory of the function f(s) was made by Cahen² in a memoir which, although much of the analysis which it contains is open to serious criticism, has served—and possibly just for that reason—as the starting point of most of the later researches in the subject³.
It is clear that all but a finite number of the numbers λn must be positive. It is often convenient to suppose that they are all positive, or at any rate that λ1 ≧ 0.⁴
2. It will be convenient at this point to fix certain notations which we shall regard as stereotyped throughout the tract.
(i) By [x] we mean the algebraically greatest integer not greater than x. By
we mean the sum of all values of f(n) for which α ≦ n ≦ β, i.e. for [α] ≦ n ≦ [β] or [α] < n ≦ [β], according as a is or is not an integer. We shall also write
⁵
(ii) We shall follow Landau in his use of the symbols o, O.⁶ That is to say, if ɸ is a positive function of a variable which tends to a limit, we shall write
f = o (ɸ)
if f/ɸ → 0, and
f = O (ɸ)
if |f|/ɸ remains less than a constant K. We shall use the letter K to denote an unspecified constant, not always the same⁷.
II
ELEMENTARY THEORY OF THE CONVERGENCE OF DIRICHLET’S SERIES
1. Two fundamental lemmas. Much of our argument will be based upon the two lemmas which follow.
LEMMA 1. We have identically
This is Abel’s classical lemma on partial summation⁸.
LEMMA 2. If σ ≠ 0, then
⁹ For
2. Fundamental Theorems. Region of convergence, analytical character, and uniqueness of the series. We are now in a position to prove the most important theorems in the elementary theory of Dirichlet’s series.
THEOREM 1. If the series is convergent for s = σ + ti, then it is convergent for any value of s whose real part is greater than σ.
This theorem is included in the more general and less elementary theorem which follows.
THEOREM 2. If the series is convergent for s = s0, then it is uniformly convergent throughout the angular region in the plane of s defined by the inequality
¹⁰
We may clearly suppose s0 = 0 without loss of generality. We have
by Lemma 1. If ∊ is assigned we can choose m0 so that λm > 0 and
for v ≧ m ≧ m0. If now we apply Lemma 2, and observe that
| s |/σ≦ sec a
throughout the region which we are considering, we obtain
for n ≧ m ≧ m0. Thus Theorem 2 is proved¹¹, and Theorem 1 is an obvious corollary.
There are now three possibilities as regards the convergence of the series. It may converge for all, or no, or some values of s. In the last case it follows from Theorem 1, by a classical argument, that we can find a number σ0 such that the series is convergent for σ > σ0 and divergent or oscillatory for σ < σ0.
THEOREM 3. The series may be convergent for all values of s, or for none, or for some only. In the last case there is a number σ0 such that the series is convergent for σ > σ0 and divergent or oscillatory for σ < σ0.
In other words the region of convergence is a half-planet ¹². We shall call σ0 the abscissa of convergence, and the line σ = σ0 the line of convergence. It is convenient to write σ0 = − ∞ or σ0 = ∞ when the series is convergent for all or no values of s. On the line of convergence the question of the convergence of the series remains open, and requires considerations of a much more delicate character.
3. Examples. (i) The series ∑ ann−s, where | a | < 1, is convergent for all values of s.
(ii) The series ∑ ann−s, where | a | > 1, is convergent for no values of s.
(iii) The series ∑n−s has σ = 1 as its line